{"id":"742d12a5-19ea-436b-ae11-6f23110c7451","arxiv_id":"2507.12647","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Bayesian platform trial power and error rates can be estimated accurately from simulations at only two interim sample sizes by exploiting an approximate linear trend in logits of posterior probabilities.","lead":"Bayesian platform trials for testing several treatments at once are expensive to design because regulators require extensive Monte Carlo simulations for every candidate sample size. This paper shows that running those simulations at just two sample sizes, then interpolating using a proven linear trend, gives accurate operating characteristics, and it applies the approach to the SSTARLET tuberculosis trial.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's constant limiting derivative does not prove finite-sample linearity of logit quantiles, so Algorithm 1's two-point interpolation and the claimed generality of the method rest on an unvalidated approximation.","rationale":"The reader identified the same load-bearing assumption: finite-sample linearity of logit quantiles in n. I agree that this is the weakest point because Theorem 1 establishes an asymptotic slope, not finite-sample straightness, and the method's central efficiency claim depends on that straightness. The empirical check in Section 5 is supportive for SSTARLET but shows small discrepancies at small sample sizes and covers only one design, so it cannot establish the broader claim. The proposed concrete test would directly measure the finite-sample error using a third set of simulation points and a second configuration, which should settle whether the approximation is reliable. If it passes, the method is sound for practical use; if it fails, the paper's generality claim needs to be weakened. I therefore move the verdict from ACCEPT to CONDITIONAL rather than REJECT, because the available evidence supports the method for the studied design and the concern is about unproven generality, not demonstrated invalidity.","tokens_in":15970,"tokens_out":7946,"duration_ms":99926,"concrete_test":"Re-run Algorithm 1 on the SSTARLET design with anchor pair na = 400 and nb = 1000, and compare the interpolated logit quantiles and the resulting power/FWER curves against direct simulation at every n ∈ {500, 600, 700, 800, 900}. Flag any deviation exceeding twice the Monte Carlo standard error of the direct estimates. Then repeat the entire comparison under one modified platform configuration, e.g., c2 = 2, a different MAP prior weight, or an additional endpoint, to test whether the method generalizes. If deviations remain within Monte Carlo noise in both configurations, the linearity concern is resolved; if not, the two-anchor method should be conditioned on per-design validation.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that the joint sampling distribution of posterior probabilities can be recovered from simulations at only two sample sizes. Algorithm 1 (Lines 8–10) linearly interpolates the logits of posterior-probability order statistics between na and nb and uses the result to estimate power and FWER at every other n. The theoretical support, Theorem 1, proves only that the derivative of the conditional logit quantiles converges to a constant as n → ∞, which is a statement about the asymptotic slope, not about the shape of the curve over a finite working range. A function whose derivative has a finite limit need not be straight; the remainder can vary with n and with the quantile level. Thus the two-point interpolation is a heuristic approximation whose finite-sample error is not bounded by the theorem. The paper's own validation detects discrepancies for Trt3 at smaller sample sizes (Figure 3B and 3D), and the recommended n1 = 674 lies between the anchors na = 600 and nb = 1000, so mild curvature within that interval matters. Only one real design (SSTARLET) is examined, so the general statement that 'only two sample sizes' suffice for Bayesian platform trials is not established by either proof or evidence. A secondary, related concern is that pairing the d-th order statistics of each component across na and nb imposes a rank-aligned joint dependence; Theorem 1 concerns conditional quantiles, not the stability of the copula, so this pairing could misrepresent the joint distribution even when each marginal is linear.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an efficient sample size determination (SSD) method for Bayesian platform trials. The method simulates the joint sampling distribution of posterior probabilities at only two interim sample sizes, then linearly interpolates the logits of posterior-probability order statistics to obtain operating characteristics at other sample sizes. The authors prove an asymptotic result (Theorem 1) on the limiting derivative of conditional logit quantiles, present Algorithm 1 as the SSD procedure, and demonstrate the approach on the SSTARLET tuberculosis trial, a realistic platform design with arm addition, a fixed outcome delay, variable allocation, and robust MAP priors.","tokens_in":16267,"tokens_out":10138,"duration_ms":118685,"significance":"If the proposed method generalizes as claimed, it would reduce the computational burden of calibrating Bayesian platform trials by replacing full-simulation grids with simulations at two sample sizes. The paper includes a substantial real-world case study, reports strong agreement between the SSD estimates and direct simulations in Table 3, and provides publicly available code, which are notable strengths. However, the theoretical support is asymptotic rather than finite-sample, and the empirical validation is limited to a single platform design, so the breadth of the central claim exceeds what is currently demonstrated.","major_comments":[{"comment":"Theorem 1 proves that the limiting derivative of the conditional logit quantiles of the proxy sampling distribution is constant as n tends to infinity. This does not establish that the logit quantiles are linear in n over a finite working range, nor does it provide an error bound for the two-point interpolation in Algorithm 1 (Lines 8-10). The manuscript itself reports discrepancies at smaller sample sizes in Figure 3B and 3D, and the simulation study considers only the SSTARLET design. The abstract's statement that the method models the joint sampling distribution 'using simulations conducted at only two sample sizes' for Bayesian platform trials is therefore stronger than what the theorem and the evidence support. I recommend either tempering the generality claims and explicitly stating that finite-sample accuracy is an empirical assumption to be verified for each design, or adding corroborating simulation studies for platform trials with different structures (for example, no outcome lag, a different number of arms, or different endpoint types) together with a discussion of conditions under which the approximation is expected to fail.","section":"§4.1, Theorem 1 and Algorithm 1"},{"comment":"The rank-aligned pairing of the d-th order statistics across components imposes a particular copula between components at the interpolated sample size. Theorem 1 concerns conditional quantiles of the proxy distribution, not the invariance of the joint dependence structure in n, so this pairing is not theoretically justified by the theorem. The issue is load-bearing for the FWER in Equation (4), which depends on the joint distribution over arms. The authors should either provide a sensitivity analysis of this copula assumption or clearly acknowledge that the joint dependence is approximated by the rank-aligned construction.","section":"Algorithm 1, Lines 6-10"}],"minor_comments":[{"comment":"The sentence 'We set w = 0.5' should be 'We set w1 = 0.5' to match the notation used in the robust MAP prior specification.","section":"§3.3"},{"comment":"The simulated curves would be easier to assess if they included pointwise Monte Carlo confidence bands or standard error shading; as presented, it is hard to separate approximation error from simulation noise.","section":"Figures 1-3"},{"comment":"The caption should state explicitly that the 'Estimated' rows use Algorithm 1 with R = 10,000 replicates and the same decision thresholds as the 'Simulated' rows, so that the comparison is unambiguous.","section":"Table 3"},{"comment":"The phrase 'prove theoretical results that allow us to model' in the abstract and the word 'guarantees' in the text are stronger than the asymptotic result supports; replacing them with 'motivates' or 'provides asymptotic support for' would better reflect the finite-sample role of the empirical validation.","section":"§4.1"},{"comment":"The proof of Theorem 1 is deferred to the online supplement, which is not included in the arXiv submission; please ensure the supplement is available to reviewers and consider including a proof sketch in the main text.","section":"Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is well written and the SSTARLET application is valuable. The main issue is the gap between the asymptotic theory and the general claim that two sample sizes suffice for Bayesian platform trials; this is addressable by softening the claims or adding further validation. I would not reject the paper, but I would want to see the generality concern addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper does a real, useful thing. It extends the two-sample-size linear-logit approximation from Hagar et al. (2025) to arbitrary linear sample-size functions, which is exactly what you need for platform trials with a fixed outcome lag and post-interim reallocation. The SSTARLET application is not a toy: robust MAP priors, mid-trial arm addition, reallocation after dropping, and they ship code. The empirical validation is the core of the paper and it holds up. Sixteen composite scenarios, R=10,000, and the SSD-based curves track the full-simulation grid well across the operating range. At the recommended n1=674, the estimated and simulated non-inferiority probabilities are within a couple of thousandths in most settings. That is convincing evidence that the approximation is good enough for design work.\n\nThe soft spots are real but not disqualifying. Theorem 1 proves that the limiting derivative of the logit quantile is constant, which does not by itself guarantee that two-point linear interpolation is accurate at finite n. The paper does not oversell this; it calls the linearity a good global approximation for large n and locally suitable for smaller n. The check on SSTARLET is one design, and the discrepancies in Figure 3B and 3D at small n are acknowledged. The appendix says those improve with more posterior draws, which points to Monte Carlo noise in the anchors rather than curvature, but the evidence is not systematic. The rank-aligned pairing of order statistics across na and nb is a heuristic for the joint distribution; Theorem 1 concerns marginal conditional quantiles, so the copula stability is not directly justified. Again, the simulation study exercises the joint distribution, so this is a gap in theory more than a demonstrated failure.\n\nMy main criticism of presentation: the abstract says 'prove theoretical results that allow us to model the joint sampling distribution... using simulations conducted at only two sample sizes.' That overstates the role of the theorem. What is proved is an asymptotic slope property; what makes the method work is the empirical linearity in this class of designs.\n\nWho is this for: statisticians designing Bayesian platform trials, especially with logistics delays. It will be useful to anyone doing simulation-calibrated Bayesian design outside the simplest settings. Give it a proper peer review. The theory is not deep enough to carry the paper alone, but the application, the validation, and the code make it a worthwhile contribution.","headline":"Solid extension of the two-anchor SSD trick to realistic platform trial constraints, honestly validated on SSTARLET; the asymptotic theorem is motivation rather than proof of finite-sample accuracy, but the empirical case is strong enough to warrant serious review.","tokens_in":16798,"tokens_out":2199,"would_cite":true,"duration_ms":24218,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62L05","62P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Simulating a Bayesian platform trial at just two interim sample sizes can recover its power and family-wise error curves, the paper claims.","keywords":["Bayesian platform trials","sample size determination","posterior probabilities","family-wise error rate","interim analysis","response-adaptive randomization","MAP priors","tuberculosis preventive therapy"],"falsifier":"Pick an intermediate sample size between the two anchors (for SSTARLET, say $n_1 = 800$ between 600 and 1,000), run the full Monte Carlo simulation of the joint sampling distribution, and compare the simulated power and family-wise error against the interpolated values at that point. Repeating this for a scenario in which an arm is barely acceptable or the null lies at the margin, where discrepancies were most visible in the paper, would settle how much curvature the linearity assumption can tolerate.","tokens_in":15778,"feed_emoji":"🩺","tokens_out":9077,"duration_ms":100119,"temperature":0.7,"pith_summary":"Bayesian platform trials, which test several treatments against a shared control while adding and dropping arms, must be checked by simulation before launch, and regulators expect those operating characteristics to be reported. Rerunning the simulation at every candidate sample size is slow when interim decisions are complex. The paper claims that the joint sampling distribution of the posterior probabilities driving those decisions has quantiles whose log-odds are nearly linear in the interim sample size, and proves this for a large-sample proxy distribution. That justifies simulating at just two anchor sample sizes and interpolating in between. Applied to the SSTARLET tuberculosis trial, the method selects an interim sample size of 674 and a final size of 1,685, matching full-grid simulations while cutting design computation from hours to minutes.","feed_headline":"Two simulated sample sizes set a Bayesian trial's size","feed_subtitle":"The paper's two-point interpolation cuts design simulation from hours to minutes while holding power and error controls.","key_machinery":"The load-bearing mechanism is two-anchor interpolation of the logits of posterior-probability quantiles: simulate synthetic trials at two interim sample sizes $n_a$ and $n_b$, compute all posterior probabilities, sort their logits, pair the order statistics, and linearly interpolate or extrapolate to any requested $n$. Theorem 1 justifies this by showing that in the proxy distribution formed by substituting asymptotically normal maximum-likelihood estimates into a Bernstein-von Mises posterior, every conditional quantile's logit has a constant limiting slope in $n$, equal to $(0.5 - I\\{\\delta^*_i \\ge \\delta_{m,i}\\})(\\delta_{m,i} - \\delta^*_i)^2 / \\Sigma_{i,i}$. Because the slope is independent of the quantile level, interpolation of the marginal quantiles also tracks the conditional structure when the replicate order statistics are paired by rank. The algorithm enumerates all possible active arm sets, evaluates power and family-wise error from the interpolated joint distribution, and finds the smallest $n$ that meets the power constraint.","core_discovery":"The paper establishes a theorem: under the regularity conditions for maximum-likelihood normality and the Bernstein-von Mises theorem, the limiting derivative of the logit of each conditional quantile of the posterior-probability sampling distribution with respect to the interim sample size is a constant that does not depend on the sample size or the quantile level. Those logits can therefore be treated as linear functions of the sample size, so the true joint sampling distribution can be estimated by Monte Carlo at two anchor sizes and linearly interpolated. The implementation pairs the order statistics of the logits at the two anchors, which preserves the dependence across endpoints, trial stages, and the possible sets of active arms even when the active set changes with the sample size. In the SSTARLET design this reproduces the simulated power and family-wise error curves across all four outcome scenarios and recommends $n_1 = 674$, $n_2 = 1{,}685$.","pith_inferences":["An implication left implicit is that the two-anchor scheme should extend recursively to trials with more than two interim analyses, with each analysis pair interpolated in turn; that extension would need its own linearity check at every window.","The rank-pairing mechanism suggests the method could transfer beyond binomial outcomes to endpoints whose posterior summaries are approximately normal, such as time-to-event or continuous endpoints, provided a suitable variance proxy exists.","The paper selects several decision thresholds heuristically; a direct follow-up would optimize all thresholds jointly against the interpolated operating-characteristic surfaces instead of tuning only the final non-inferiority threshold.","A low-cost robustness check, not pursued in the paper, is to add a third anchor sample size whenever prior information or boundary-null scenarios are expected to bend the quantile curves, turning the linear interpolation into a diagnostic."],"forward_implications":["A platform trial with the same structure can be re-evaluated for different thresholds or sample sizes without new simulations: the two anchor distributions are reused and only the interpolation target changes.","Power and family-wise error control are approximately preserved across the whole examined sample-size range, so the recommended sample size and thresholds come with operating-characteristic curves rather than a single point estimate.","The framework extends to other platform designs as long as each arm's stage-wise sample sizes are linear functions of the interim sample size, a condition the paper states as a prerequisite.","When the null model sits exactly at the non-inferiority margin, the limiting slopes vanish, so thresholds chosen at one anchor approximately maintain the family-wise error rate at all sample sizes."],"supporting_citations":[{"why":"Supplies the regularity conditions and asymptotic normality/Bernstein-von Mises results that define the proxy sampling distribution and prove Theorem 1.","marker":"van der Vaart (1998)"},{"why":"The previous sequential-design framework being extended; its assumption of proportional sample sizes fails for platform trials with delayed decisions.","marker":"Hagar et al. (2025)"},{"why":"Establishes the two-sample-size strategy for estimating sampling distributions of posterior summaries in non-sequential settings.","marker":"Hagar and Stevens (2025)"},{"why":"Extends the sample-size-space approach to clustered data and multiple endpoints, leaving the platform-trial gap this paper fills.","marker":"Hagar and Golchi (2025)"},{"why":"Proves that the proxy sampling distribution converges in total variation to the true sampling distribution as sample size grows.","marker":"Hagar and Stevens (2024)"},{"why":"Supports the variance structure linking estimands across trial stages in the proxy distribution.","marker":"Jennison and Turnbull (2000)"},{"why":"Provides the robust meta-analytic-predictive prior construction and the recommended mixture weight used for historical borrowing.","marker":"Schmidli et al. (2014)"},{"why":"Supplies the historical adverse-event data used to construct informative priors for the 2R20 and 4R10 arms.","marker":"Ruslami et al. (2024)"}],"fun_headline_variants":["Two simulation points size Bayesian platform trials","Bayesian trial design from only two Monte Carlo runs","Linearity theorem cuts platform trial simulation load","SSTARLET trial sized via two-point interpolation","Efficient Bayesian platform trial design with two anchors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method rests on assuming that the log-odds (logit) of each posterior-probability quantile is nearly a straight line in the sample size, so interpolation between just two simulated sizes tracks the true curves; if those curves bend, the resulting power and error-rate estimates drift.","fun_headline_variants_meta":{"raw":{"variants":["Two simulation points size Bayesian platform trials","Bayesian trial design from only two Monte Carlo runs","Linearity theorem cuts platform trial simulation load","SSTARLET trial sized via two-point interpolation","Efficient Bayesian platform trial design with two anchors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1556,"prompt_tokens":923,"completion_tokens":633,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":563}},"tokens_in":539,"tokens_out":633,"duration_ms":7922,"temperature":1.0,"reasoning_tokens":563,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:43:55.477627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick an intermediate sample size between the two anchors (for SSTARLET, say $n_1 = 800$ between 600 and 1,000), run the full Monte Carlo simulation of the joint sampling distribution, and compare the simulated power and family-wise error against the interpolated values at that point. Repeating this for a scenario in which an arm is barely acceptable or the null lies at the margin, where discrepancies were most visible in the paper, would settle how much curvature the linearity assumption can tolerate.","supporting_citations":[{"cited_title":"and Stevens, N","cited_arxiv_id":null,"evidence_quote":"Proves that the proxy sampling distribution converges in total variation to the true sampling distribution as sample size grows."},{"cited_title":"and Turnbull, B","cited_arxiv_id":null,"evidence_quote":"Supports the variance structure linking estimands across trial stages in the proxy distribution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the robust meta-analytic-predictive prior construction and the recommended mixture weight used for historical borrowing."},{"cited_title":"J., Fisher, D., Flores, E., Fox, G","cited_arxiv_id":null,"evidence_quote":"Supplies the historical adverse-event data used to construct informative priors for the 2R20 and 4R10 arms."}],"review_version":1}